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Theorem dp15lemh 1159
 Description: Part of proof (1)=>(5) in Day/Pickering 1982. (Contributed by NM, 2-Apr-2012.)
Hypotheses
Ref Expression
dp15lema.1 d = (a2 ∪ (a0 ∩ (a1b1)))
dp15lema.2 p0 = ((a1b1) ∩ (a2b2))
dp15lema.3 e = (b0 ∩ (a0p0))
dp15lemg.4 c0 = ((a1a2) ∩ (b1b2))
dp15lemg.5 c1 = ((a0a2) ∩ (b0b2))
Assertion
Ref Expression
dp15lemh ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((c0c1) ∪ (b1 ∩ (a0a1)))

Proof of Theorem dp15lemh
StepHypRef Expression
1 dp15lema.1 . . . . . 6 d = (a2 ∪ (a0 ∩ (a1b1)))
2 dp15lema.2 . . . . . 6 p0 = ((a1b1) ∩ (a2b2))
3 dp15lema.3 . . . . . 6 e = (b0 ∩ (a0p0))
41, 2, 3dp15lemc 1154 . . . . 5 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ (((a0 ∪ (a2 ∪ (a0 ∩ (a1b1)))) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ ((a1 ∪ (a2 ∪ (a0 ∩ (a1b1)))) ∩ (b1b2)))
51, 2, 3dp15lemd 1155 . . . . 5 (((a0 ∪ (a2 ∪ (a0 ∩ (a1b1)))) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ ((a1 ∪ (a2 ∪ (a0 ∩ (a1b1)))) ∩ (b1b2))) = (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2)))
64, 5lbtr 139 . . . 4 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2)))
71, 2, 3dp15leme 1156 . . . 4 (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (a0 ∩ (a1b1))) ∩ (b1b2))) ≤ (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)))
86, 7letr 137 . . 3 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2)))
91, 2, 3dp15lemf 1157 . . 3 (((a0a2) ∩ ((b0 ∩ (a0p0)) ∪ b2)) ∪ (((a1a2) ∪ (b1 ∩ (a0a1))) ∩ (b1b2))) ≤ (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1))))
108, 9letr 137 . 2 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1))))
11 dp15lemg.4 . . 3 c0 = ((a1a2) ∩ (b1b2))
12 dp15lemg.5 . . 3 c1 = ((a0a2) ∩ (b0b2))
131, 2, 3, 11, 12dp15lemg 1158 . 2 (((a1a2) ∩ (b1b2)) ∪ (((a0a2) ∩ (b0b2)) ∪ (b1 ∩ (a0a1)))) = ((c0c1) ∪ (b1 ∩ (a0a1)))
1410, 13lbtr 139 1 ((a0a1) ∩ ((b0 ∩ (a0p0)) ∪ b1)) ≤ ((c0c1) ∪ (b1 ∩ (a0a1)))
 Colors of variables: term Syntax hints:   = wb 1   ≤ wle 2   ∪ wo 6   ∩ wa 7 This theorem was proved from axioms:  ax-a1 30  ax-a2 31  ax-a3 32  ax-a4 33  ax-a5 34  ax-r1 35  ax-r2 36  ax-r4 37  ax-r5 38  ax-ml 1120  ax-arg 1151 This theorem depends on definitions:  df-a 40  df-t 41  df-f 42  df-le1 130  df-le2 131 This theorem is referenced by:  dp15  1160
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